GCSE Maths Practice: standard-form

Question 6 of 10

Practise converting standard form expressions into ordinary numbers using powers of ten.

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\( \begin{array}{l}\text{What is } 3.5 \times 10^3 \text{ in ordinary number form?}\end{array} \)

Choose one option:

Each increase in exponent adds a zero. For example, 10³ = 1000, so multiply the coefficient by 1000.

Converting from Standard Form to Ordinary Numbers

In GCSE Maths, it is just as important to convert numbers from standard form back into ordinary form as it is to convert the other way. Standard form expresses a number as a value between 1 and 10 multiplied by a power of ten. When converting back, you simply reverse that process by expanding the power of ten into full numerical form. This skill helps you move easily between compact scientific notation and everyday decimal numbers.

Understanding Positive Powers

A positive power of ten means that the number is greater than one. Each increase in the exponent represents one extra zero added to the value when written in full. For example, multiplying by 10³ means multiplying by one thousand. In other words, you move the decimal point three places to the right. This makes small decimals grow into whole or larger numbers.

Step-by-Step Method

  1. Write down the number that appears before the multiplication sign (the coefficient).
  2. Identify the power of ten.
  3. Move the decimal point to the right if the power is positive, or to the left if it is negative.
  4. Fill in any empty spaces with zeros as placeholders.

Worked Example 1

Convert 4.2 × 10² into ordinary form.

  • Power of ten = 2, so move the decimal two places right.
  • Result: 420.

Worked Example 2

Convert 7.08 × 10⁴ into ordinary form.

  • Move the decimal four places right.
  • Result: 70,800.

Worked Example 3

Convert 6.1 × 10⁵ into ordinary form.

  • Move the decimal five places right.
  • Result: 610,000.

Common Mistakes

  • Moving the decimal the wrong way. Remember: positive power → right, negative power → left.
  • Forgetting to insert zeros when there are not enough digits.
  • Misreading the exponent; 10³ and 10⁵ differ by a factor of one hundred.

Real-Life Connections

Large numbers written in standard form often appear in science, geography, and finance. For example, Earth’s population (~8.1 × 10⁹) can be written as 8,100,000,000 in ordinary form. Engineers working with electrical circuits convert small currents like 4.5 × 10⁻³ A into ordinary form when checking precise readings. Understanding this translation ensures accurate interpretation in real-world contexts.

FAQs

  • What if the power is zero? Any number multiplied by 10⁰ stays the same, because 10⁰ = 1.
  • Can the coefficient be more than one digit? In standard form, the coefficient must always be between 1 and 10. If not, adjust the decimal and power.
  • Do I always move the decimal right for positive powers? Yes — positive powers make numbers larger, so move the decimal to the right.

Study Tip

When you see a positive exponent, picture the decimal ‘marching’ right across the digits, leaving zeros behind it. Practising with both positive and negative powers builds confidence for all types of standard form conversions on your GCSE Maths exam.